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366                               CHEMICAL ENGINEERING
average of the difference between the temperatures prevailing inside the evaporators and the room. This loss by radiation should be added to the total amount required for evaporation.
The equation for steam consumption may now be rewritten as follows:
S  Fi + W/L(ti - t)
where, $ denotes pounds of steam entering first steam chest,
Vi denotes pounds of evaporation, consisting of difference in weight between ingoing and outgoing liquor and loss by radiation multiplied by factor of distribution, IF denotes pounds of liquor entering the evaporator, L denotes latent heat of steam entering steam chest, ti denotes temperature of boiling liquid, and t denotes temperature of liquor entering evaporator.
The concentration of 14,000 Ib. of weak liquor at 70F. to 2,000 Ib. in a triple effect with steam at 5 Ib. pressure and a vacuum of about 27 in. in the last effect, would require an evaporator with an outside surface of about 3 X 800 = 2,400 sq. ft., and the loss by radiation would be R = 2,400(160 - 70) X 0.5/960 = 113 Ib., if 160 is the average temperature in the three evaporators. The temperature in the first effect would be 190F., and
7i = (14,000 Ib. - 2,000 Ib. -f 113 Ib.) X 0.295 S = 3,573 + 14.6(190 - 70)    = 5,325 Ib.
If the weak liquor should have a temperature of 212 instead of 70, then S = 3,573 + 14.4(190 - 212) - 3,541 Ib.
which shows plainly the importance of the temperature of the ingoing liquor. In cases where the liquors contain large percentages of solids these should be taken into account when figuring the steam required for pre-heating.
The foregoing figures have all been based on the usual practice of running the multiple effect "direct current," which means that the weak liquor enters the first effect and passes from there to the second, third and fourth effect. In some cases, however, it has been found advantageous to run "countercurrent" by feeding the liquor into the last effect and convey it from there by pumping to the previous evaporators. Naturally there will be no reevaporation and the proportion of vapors will be distributed so that the first effect will have the highest and the last evaporator the lowest amount of evaporation, as the liquor has to be heated going from one effect to the other. This method of running countercurrent will save a small percentage of heating steam in case large amounts of cold liquors are entering the evaporating system, and has also other advantages: It will eliminate losses by entrainment which is frequently caused by the rapid reevaporation of the hot liquor entering the next evaporator; it will save from 20 to 40 per cent of cooling water; it will boil and discharge the finished liquor at high temperature. Countercurrent operation will require, however, pumping of the liquor from one effect to the next, and this is frequently undesirable especially with chemical liquors that will corrode metals. Further details may be found in a paper read by H. K. Moore and published in "Chemical and Metallurgical Engineering," Vol. 18, p. 187 et seq.
A third method is "semi-countercurrent" where the weak liquor enters the second effect and passes from there to the third and fourth. The liquor of the last evaporator is pumped into the first, and discharged at high temperature (black liquors and packing-house tank waters).
In some industries, particularly in beet-sugar factories, vapor is taken from the first or second effect to preheat the raw juice, and this will naturally introduce complications in the calculation of the total steam consumption (see E. Hausbrand).